On change detection in a Kalman filter based tracking problem
Introduction
The Kalman filter is a strong candidate for tracking mobile nodes in wireless networks [1]. However, mobile nodes may suddenly change their movement pattern while being tracked. The Kalman filter in its basic form will rarely detect such changes quickly; initially the filter will treat such events as spurious noise, rather than a change in the state. The present paper focuses on the detection of impulsive state changes.
Kalman filtering in the presence of changes to the plant and observation equations has been studied during the past decades [2], [3] and still attracts interests in new areas [4], [5]. The source of the change can be a multitude of causes that distorts the expected evolution, and it can therefore be modeled in different ways (e.g. in state evolution matrix or an additive change vector). In order to capture such changes, different solutions are suggested. In [3], and the references therein, the focus is confined to various hybrid estimation approaches. Briefly, hybrid estimation is the estimation of the state vector that has both continuous and discrete components. A wide variety of solutions have been proposed for these types of problems. For example, an adaptive neural controller is suggested in [6] for a nonlinear tracking system and in [7] authors proposed a fuzzy inference system to distinguish between noisy and noise-less pixels in order to remove artifact from images. However, in this work, we confine the study to classical detectors. Which means that, it is first decided if there has been a change in the model and then the state is estimated accordingly. The literature on change detection is rich, and selected works include [2], [8].
In the present paper, and in order to detect sudden state changes, we choose to modify the standard plant equation. Specifically, we add an extra term which can account for the change. This modification makes it possible, for example, to recognize the movement of a node which stops and starts abruptly. Such changing movement patterns are basic in many widely used movement models, such as random way point (RWP) [9]. The most common method for change detection in the Kalman-based tracking problem is to monitor the energy of the innovation. Once it goes beyond a certain level, one decides that a change has happened. The analysis in [2] shows that by incorporating prior knowledge about the nature of the change, the detection performance can be improved significantly.
The present paper studies detection of additive change. The main contributions are as follows. Firstly, we give a closed form expression for the change signature. That is, the extra term added to the innovation whenever there is a change. We also present the convergence properties and the impulse response of the change signature. Secondly, assuming impulsive change, where a single realization of a Gaussian random vector is added to the plant equation, we present a closed form approximation of the distribution of the Neyman–Pearson detector. In the asymptotic case, when the Gaussian random vector has infinite variance, we show that this detector has no benefit from knowing that the change is impulsive. Instead, it is better to invoke a Generalized Likelihood Ratio Test (GLRT), which neglects the prior information.
The rest of the paper is organized as follows. Section 2 introduces the system model, recalls the standard Kalman filter equations and derives a closed form expression for the change signature. In Section 3 the detection problem is investigated under different levels of prior knowledge about the change. In Sections 4 and 5, some numerical examples are studied, and Section 6 concludes the paper.
Section snippets
System model
We assume a linear time invariant (LTI) stochastic state-space model where the state and the observation evolve according to the following equations:Here, , are the state and the observation vectors respectively, at time instant k. The vectors and are the process and observation noise respectively, which are assumed to be mutually independent and white, with covariance matrix and respectively. The matrices and are the state
The detection problem
Using the results from the previous we can now formulate a signal detection problem. We treat (12) as the observation, ek0 as the observation noise and as the desired signal to be detected. Clearly, any detectable change will manifest itself through a non-zero . The two hypotheses of the detection problem can therefore be stated as follows:In other words, under there is no change, whereas under there is a change. For the rest of this paper, we will assume v
Detection performance
Here we consider a numerical example, and study both the change signature and detection performance. We assume a tracking scenario where the target moves along the x-axis with a constant speed. Similar to [15], we assume the following model parameters in (1):where t, xk and vk are the sampling interval, location on the x-axis and the speed, both at time k, respectively. It is assumed that after reaching the destination the target stops and its speed drops to
Performance evaluation in a tracking scenario
As a practical example, we consider a tracking problem where the target movements consist of many straight line branches. For each branch, the target speed, v, is assumed to be a Gaussian Random variable . For simplicity it is assumed that the target moves only in x and y directions (i.e. on z=0 plane) and this movement is observed using 16 sensors which are located on a parallel plane at . It is assumed that the target stops at the end of each branch for a random amount of
Conclusion
We have studied additive change detection for a Kalman-based tracking problem, and expressed in closed form the effect of the additive change on the innovation. Emphasizing on detection of impulsive changes, we have derived detectors for different levels of prior knowledge on the change vector. The clear trend is that the detection performance increases with increased knowledge. For the case when the change vector is assumed to be Gaussian, and its variance tends to infinity, the analysis has
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